Problem 22 - Calculus for Biology and Medicine

Question

Forest Succession Forest succession can be modeled by a three-compartment model. We assume that gaps in a forest are created by disturbances just as in Problem 21 . These gaps are initially filled by fast-growing, early colonizing plants, which are then replaced by slower growing species, a process known as succession. We denote by \(x_{1}(t)\) the total area occupied by gaps at time \(t\), by \(x_{2}(t)\) the total area occupied by fast growing species at time \(t\), and by \(x_{3}(t)\) the total area occupied by slow growing species at time \(t\). The dynamics are given by $$ \begin{array}{l} \frac{d x_{1}}{d t}=0.2 x_{2}+x_{3}-2 x_{1} \\ \frac{d x_{2}}{d t}=2 x_{1}-0.7 x_{2} \\ \frac{d x_{3}}{d t}=0.5 x_{2}-x_{3} \end{array} $$ (a) Draw the corresponding compartment diagram. (b) Show that $$ x_{1}(t)+x_{2}(t)+x_{3}(t)=A $$ where \(A\) is a constant, and give the meaning of \(A\).

Step-by-Step Solution

Answer

A is the total forest area, constant over time.

Step 1: Understanding the Compartment Diagram

The model describes three compartments representing different forest areas: gaps, fast-growing plants, and slow-growing plants. Each compartment has flows that represent changes over time. These flows are based on the given differential equations. For instance, the flow to gaps is influenced by disturbances, and it decreases as gaps are filled by fast-growing plants.

Step 2: Drawing the Compartment Diagram

From the given differential equations, draw arrows representing flows between the compartments. An arrow from fast-growing species to gaps indicates the rate 0.2. An arrow from slow-growing species to gaps indicates a factor of 1. An arrow from gaps to fast-growing species indicates a rate of 2. Finally, an arrow from fast-growing to slow-growing species represents a factor of 0.5. Negative coefficients represent outflows from compartments.

Step 3: Understanding the Constant A

The constant A represents the total area of the forest, which is the sum of all areas across compartments, assumed unchanged over time. This means no area is gained or lost, merely transformed between states.

Step 4: Showing Constant Total Area

Differentiate the total area with respect to time: \(\frac{d}{dt}[x_{1}(t)+x_{2}(t)+x_{3}(t)] = \frac{d x_{1}}{d t} + \frac{d x_{2}}{d t} + \frac{d x_{3}}{d t}\). Substitute from the given equations: \(\frac{d x_{1}}{d t} = 0.2x_2 + x_3 - 2x_1\), \(\frac{d x_{2}}{d t} = 2x_1 - 0.7x_2\), \(\frac{d x_{3}}{d t} = 0.5x_2 - x_3\). Sum these: \(\frac{d}{dt}[x_{1} + x_{2} + x_{3}]\) simplifies to zero, showing the total remains constant.

Key Concepts

Three-Compartment Model

Forest succession illustrates the cycle of change in forest ecosystems over time. To model this, we can use a three-compartment model. The compartments represent different parts of the forest: gaps, fast-growing species, and slow-growing species. The three-compartment model helps us understand how each part changes and interacts with each other.

• First, you have **gaps** in the forest caused by disturbances like fires or logging. This is where regeneration starts.
• Next, **fast-growing species** initially colonize these gaps and rapidly cover them. These plants thrive in open spaces.
• Lastly, **slow-growing species** eventually take over as they are more stable and sustainable in the long term.

The model shows how these processes occur through linear flows, which means changes are proportional to the quantities involved. This simplified view allows ecologists to predict changes under different circumstances.

Differential Equations

In order to predict changes in each compartment over time, we use differential equations. These equations describe the rates at which each area changes.
For example, the rate of change of gaps, fast, and slow-growing species are expressed mathematically:
\[\frac{dx_1}{dt} = 0.2x_2 + x_3 - 2x_1, \\frac{dx_2}{dt} = 2x_1 - 0.7x_2, \\frac{dx_3}{dt} = 0.5x_2 - x_3\]

• **First equation**: The change in gaps (9x_1/dt) is influenced by occupancy by other species.
• **Second equation**: The growth of fast-growing species (9x_2/dt) is mostly determined by the conversion of gaps to areas they can occupy.
• **Third equation**: The expansion of slow-growing species (9x_3/dt) is typically from succession from fast to slow-growing species.

These equations help ecologists understand what factors most affect changes in forest composition over time.

Compartment Diagram

A compartment diagram visually represents the relationships and changes between different compartments. This helps in easily understanding the direction and magnitude of flows between them.
To draw a compartment diagram for forest succession:

• Use arrows to denote the direction of the flow between compartments.
• The **arrows** are labeled with the rate factors from the differential equations.
• Positive flow values indicate addition, whereas negative values show outflows.

For this exercise:

• An arrow from **fast-growing species to gaps**, representing the interactions and rates, is marked by a factor of **0.2**.
• There is another arrow from **slow-growing plants to gaps** with a factor of **1**.
• Arrows can also show outflows, such as fast-growing to slow-growing species, marked as **0.5**.

The compartment diagram is akin to a map of ecological processes, making complex relationships easier to digest at a glance.

Ecological Modeling

Ecological models, such as this three-compartment model, are essential tools in understanding and predicting environmental changes.
These models offer insights into:

• **Forest dynamics**, by showing how different components interact and change over time.
• The impact of **disturbances**, such as natural calamities or human activities, which create gaps.
• How species **succession** progresses by different rates of growth and replacement.

The three-compartment model simplifies complex processes so that scientists can focus on particular parts of the ecosystem dynamics. It shows that while individual components of the forest change, the **total area** remains constant, represented by a fixed constant **A** over time. This reflects the *law of conservation of area*, where nothing is lost, just transformed. Models like this help in resource management and conservation efforts by predicting future scenarios.